Slow granular flows: The dominant role of tiny fluctuations

What mechanism governs slow flows of granular media? Microscopically, the grains experience enduring frictional contacts in these flows. However, a straightforward translation to a macroscopic frictional rheology, where the shear stresses are proportional to the normal stresses with a rate-independent friction coefficient, fails to capture important aspects of slow granular flows. There is now overwhelming evidence that agitations, tiny fluctuations of the grain positions, associated with large fluctuation of their contact forces, play a central role for slow granular flows. These agitations are generated in flowing regions, but travel deep inside the quiescent zones, and lead to a nonlocal rheology.     

Debris flows: Experiments and modelling

Debris flows and debris avalanches are complex, gravity-driven currents of rock, water and sediments that can be highly mobile. This combination of component materials leads to a rich morphology and unusual dynamics, exhibiting features of both granular materials and viscous gravity currents. Although extreme events such as those at Kolka Karmadon in North Ossetia (2002) [1] and Huascarán (1970) [2] strongly motivate us to understand how such high levels of mobility can occur, smaller events are ubiquitous and capable of endangering infrastructure and life, requiring mitigation. Recent progress in modelling debris flows has seen the development of multiphase models that can start to provide clues of the origins of the unique phenomenology of debris flows. However, the spatial and temporal variations that debris flows exhibit make this task challenging and laboratory experiments, where boundary and initial conditions can be controlled and reproduced, are crucial both to validate models and to inspire new modelling approaches. This paper discusses recent laboratory experiments on debris flows and the state of the art in numerical models.     

Physics of particulate flows: From sand avalanche to active suspensions in plants

Flows of granular media in air or in a liquid have been a research field for physicists for several decades. Sometimes solid, sometimes liquid, these particulate materials exhibit peculiar behaviors, which have motivated many studies at the frontiers between nonlinear physics, soft matter physics and fluid mechanics. This paper presents a summary of the recent advances in the field, with a focus on the development of continuous approaches, which make it possible to treat granular media as a complex fluid and to develop a granular hydrodynamics. We also discuss how the better understanding of granular flows we have today may help to address more complex materials, such as colloidal suspensions or some biological systems.     

Lattice-BGK approach to simulating granular flows

Many continuum theories for granular flow produce an equation of motion for the fluctuating kinetic energy density (granular temperature) that accounts for the energy lost in inelastic collisions. Apart from the presence of an extra dissipative term, this equation is very similar in form to the usual temperature equation in hydrodynamics. It is shown how a lattice-kinetic model based on the Bhatnagar-Gross-Krook (BGK) equation that was previously derived for a miscible two-component fluid may be modified to model the continuum equations for granular flow. This is done by noting that the variable corresponding to the concentration of one species follows an equation that is essentially analogous to the granular temperature equation. A simulation of an unforced granular fluid using the modified model reproduces the phenomenon of clustering instability, namely the spontaneous agglomeration of particles into dense clusters, which occurs generically in all granular flows. The success of the continuum theory in capturing the gross features of this basic phenomenon is discussed. Some shear flow simulations are also presented.     

稠密气固流动二阶矩摩擦应力模型和模拟

基于统一二阶矩两相湍流应力模型和稠密颗粒动力学理论,建立了欧拉-欧拉双流体二阶矩颗粒摩擦应力模型。模型充分反映各向异性的气固两相相间雷诺应力相互作用,并引入有效颗粒弹性恢复系数,考虑了因颗粒表面不光滑产生的摩擦力对湍流流动结构和颗粒弥散特性的影响,对于下降管的计算结果与实验吻合较好。因颗粒摩擦产生能量耗散降低了颗粒温度和导致颗粒雷诺应力再分布。在入口和出口区域因颗粒碰撞频率较高而产生的能量耗散对流动结构影响明显。     

Coarsening in granular systems

We review a few representative examples of granular experiments or models where phase separation, accompanied by domain coarsening, is a relevant phenomenon. We first elucidate the intrinsic non-equilibrium, or athermal, nature of granular media. Thereafter, dilute systems, the so-called “granular gases”, are discussed: idealized kinetic models, such as the gas of inelastic hard spheres in the cooling regime, are the optimal playground to study the slow growth of correlated structures, e.g., shear patterns, vortices, and clusters. In fluidized experiments, liquid–gas or solid–gas separations have been observed. In the case of monolayers of particles, phase coexistence and coarsening appear in several different setups, with mechanical or electrostatic energy input. Phenomenological models describe, even quantitatively, several experimental measures, both for the coarsening dynamics and for the dynamic transition between different granular phases. The origin of the underlying bistability is in general related to negative compressibility from granular hydrodynamics computations, even if the understanding of the mechanism is far from complete. A relevant problem, with important industrial applications, is related to the demixing or segregation of mixtures, for instance in rotating tumblers or on horizontally vibrated plates. Finally, the problem of compaction of highly dense granular materials, which is relevant in many practical situations, is usually described in terms of coarsening dynamics: there, bubbles of misaligned grains evaporate, allowing the coalescence of optimally arranged islands and a progressive reduction of the total occupied volume.     

Particle-size segregation in dense granular avalanches

Particles of differing sizes are notoriously prone to segregate, which is a chronic problem in the manufacture of a wide variety of products that are used by billions of people worldwide every day. Segregation is the single most important factor in product non-uniformity, which can lead to significant handling problems as well as complete batches being discarded at huge financial loss. It is generally regarded that the most important mechanism for segregation is the combination of kinetic sieving and squeeze expulsion in shallow granular avalanches. These free-surface flows are more common than one might expect, often forming part of more complicated flows in drums, heaps and silos, where there is mass exchange with underlying regions of static or slowly moving grains. The combination of segregation and solid–fluid granular phase transitions creates incredibly complicated and beautiful patterns in the resulting deposits, but a full understanding of such effects lies beyond our capabilities at present. This paper reviews recent advances in our ability to model the basic segregation processes in a single avalanche (without mass exchange) and the subtle feedback effects that they can have on the bulk flow. This is particularly important for geophysical applications, where segregation can spontaneously self-channelize and lubricate the flow, significantly enhancing the run-out of debris-flows, pyroclastic flows, rock-falls and snow-slab avalanches.     

Kinetic Theory of Polydisperse Granular Mixtures: Influence of the Partial Temperatures on Transport Properties—A Review

It is well-recognized that granular media under rapid flow conditions can be modeled as a gas of hard spheres with inelastic collisions. At moderate densities, a fundamental basis for the determination of the granular hydrodynamics is provided by the Enskog kinetic equation conveniently adapted to account for inelastic collisions. A surprising result (compared to its molecular gas counterpart) for granular mixtures is the failure of the energy equipartition, even in homogeneous states. This means that the partial temperatures Ti (measuring the mean kinetic energy of each species) are different to the (total) granular temperature T. The goal of this paper is to provide an overview on the effect of different partial temperatures on the transport properties of the mixture. Our analysis addresses first the impact of energy nonequipartition on transport which is only due to the inelastic character of collisions. This effect (which is absent for elastic collisions) is shown to be significant in important problems in granular mixtures such as thermal diffusion segregation. Then, an independent source of energy nonequipartition due to the existence of a divergence of the flow velocity is studied. This effect (which was already analyzed in several pioneering works on dense hard-sphere molecular mixtures) affects to the bulk viscosity coefficient. Analytical (approximate) results are compared against Monte Carlo and molecular dynamics simulations, showing the reliability of kinetic theory for describing granular flows.     

Cooling kinetics of a granular gas of viscoelastic particles

The evolution of a granular gas of viscoelastic particles in the homogeneous cooling state is studied. The velocity distribution function of granular particles and the time dependence of the mean kinetic energy of particles (granular temperature) are found. The noticeable deviation of the distribution function from the Maxwell distribution and its non-monotonous evolution are established. The perturbation theory with respect to the small dispersion parameter is elaborated and the analytical expressions for the asymptotic time dependence of the velocity distribution function and the granular gas temperature are derived.     

Self-Similar Asymptotics for the Boltzmann Equation with Inelastic and Elastic Interactions

We consider some questions related to the self-similar asymptotics in the kinetic theory of both elastic and inelastic particles. In the second case we have in mind granular materials, when the model of hard spheres with inelastic collisions is replaced by a Maxwell model, characterized by a collision frequency independent of the relative speed of the colliding particles. We first discuss how to define the n-dimensional (n = 1,2,...) inelastic Maxwell model and its connection with the more basic Boltzmann equation for inelastic hard spheres. Then we consider both elastic and inelastic Maxwell models from a unified viewpoint. We prove the existence of (positive in the inelastic case) self-similar solutions with finite energy and investigate their role in large time asymptotics. It is proved that a recent conjecture by Ernst and Brito devoted to high energy tails for inelastic Maxwell particles is true for a certain class of initial data which includes Maxwellians. We also prove that the self-similar asymptotics for high energies is typical for some classes of solutions of the classical (elastic) Boltzmann equation for Maxwell molecules. New classes of (not necessarily positive) finite-energy eternal solutions of this equation are also studied.     

Cumulative author index of Volumes 351 to 360

The dynamic evolution of granular gases is fundamentally different from molecular gases due to the energy loss during collisions. Nevertheless techniques of kinetic theory are useful in a regime, when the granular particles are moving rapidly and the gas is sufficiently dilute. In these lecture notes we analyse in detail the collision of two rough particles which is inelastic due to incomplete normal and tangential restitution as well as Coulomb friction. Based on the Walton model a time evolution operator for the many particle system is introduced, a formalism which is well suited for simple approximations. We discuss free cooling of granular particles with particular emphasis on the exchange of energy between rotational and translational degrees of freedom.     

Temperature Properties in Polydisperse Granular Mixtures

Numerical simulations are employed to consider the problem of determining the granular temperatures of the species of a homogeneous heated granular mixture with a power-law size distribution. The partial granular temperature ratios are studied as functions of the fractal dimension D, the restitution coefficient e, the rescaled viscosity time, the average occupied area fraction φ, the total particle number N and the number fraction. Different species of particles in a power-law system typically do not have the same mean kinetic energy, namely the granular temperature. It is found that the extent of nonequipartition of kinetic energy is determined by the fractal dimension D, the restitution coefficient e and the rescaled viscosity time, while is insensitive to the total particle number N , the area fraction φ and the number fraction.     

数值模拟颗粒旋转对流化床内气固两相流动特性的影响

流化床内颗粒自旋转将影响颗粒相的流动特性.本文运用基于颗粒动理学理论的欧拉-欧拉气固多相流模型,考虑颗粒自旋转流动对颗粒碰撞能量交换和耗散的影响,数值模拟流化床内气体颗粒两相流动特性.计算结果表明颗粒的自旋转使得床内更容易形成气泡,颗粒浓度分布变化增大.颗粒自旋转运动将导致床内非均匀结构更明显.     

Nonlinear dynamics of density waves in granular flows through narrow vertical channels

We study granular flows through narrow channels driven by gravity in the framework of the kinetic theory for dissipative dense gases. We derive equations of motion for quasi-one-dimensional systems. In a certain range of flow density, the steady homogeneous regime is found to be unstable against the formation of density waves. We show moreover that near the onset of the instability, the governing equation for the flow density is a mixture of the Korteweg-de-Vries equation, which leads to soliton, and the Bürger equation which exhibits spatio-temporal chaos. The competition between chaos and solitons may lead either to regular spatially ordered density waves or to chaotic dynamics. We argue that these two types of dynamics can be encountered experimentally according to the channel width and the dissipative properties of the granular media. Received: 11 March 1998 / Revised and Accepted: 3 July 1998     

Kinetic Theory and Memory Effects of Homogeneous Inelastic Granular Gases under Nonlinear Drag

We study a dilute granular gas immersed in a thermal bath made of smaller particles with masses not much smaller than the granular ones in this work. Granular particles are assumed to have inelastic and hard interactions, losing energy in collisions as accounted by a constant coefficient of normal restitution. The interaction with the thermal bath is modeled by a nonlinear drag force plus a white-noise stochastic force. The kinetic theory for this system is described by an Enskog–Fokker–Planck equation for the one-particle velocity distribution function. To get explicit results of the temperature aging and steady states, Maxwellian and first Sonine approximations are developed. The latter takes into account the coupling of the excess kurtosis with the temperature. Theoretical predictions are compared with direct simulation Monte Carlo and event-driven molecular dynamics simulations. While good results for the granular temperature are obtained from the Maxwellian approximation, a much better agreement, especially as inelasticity and drag nonlinearity increase, is found when using the first Sonine approximation. The latter approximation is, additionally, crucial to account for memory effects such as Mpemba and Kovacs-like ones.     

The anisotropy of free path in a vibro-fluidized granular gas

The free path of a vibro-fluidized two-dimensional(2D) inelastic granular gas confined in a rectangular box is investigated by 2D event-driven molecular simulation. By tracking particles in the simulation, we analyze the local free path.The probability distribution of the free path shows a high tail deviating from the exponential prediction. The anisotropy of the free path is found when we separate the free path to x and y components. The probability distribution of y component is exponential, while x component has a high tail. The probability distribution of angle between the relative velocity and the unit vector joined two particle centers deviates from the distribution of two random vectors, indicating the existence of the dynamic heterogeneities in our system. We explain these results by resorting to the kinetic theory with two-peak velocity distribution. The kinetic theory agrees well with the simulation result.     

Effect of gravity on the development of homogeneous shear turbulence laden with finite-size particles

Fully resolved simulations of homogeneous shear turbulence (HST) laden with sedimenting spherical particles of finite size have been performed to clarify the effects of gravity on the development of particle-laden turbulent shear flows. We consider turbulence in a horizontal flow subjected to vertical or horizontal shear. Numerical results show that the development of HST laden with finite-size particles are significantly altered by gravity. The effects of gravity lead to a slower increase in the Taylor-microscale Reynolds number, whose value is found to be well correlated with the average particle Reynolds number. The gravity also causes a slower increase in the turbulence kinetic energy (TKE) through the enhancement of energy dissipation. The change in the Reynolds shear stress (RSS) due to particles also significantly contributes to the relative change in TKE. In vertically sheared cases, RSS has high values between counter-rotating trailing vortices behind the particles, which causes a transient relative increase in TKE. In horizontally sheared cases, on the other hand, RSS is reduced in the wakes of particles, which contributes to a significant relative reduction in TKE.     

Rheological properties of dense granular flows

Recent progresses in understanding the behavior of dense granular flows are presented. After presenting a bulk rheology of granular materials, I focus on the new developments to account for non-local effects, and on ongoing research concerning the surface rheology and the evolution of mechanical properties for heterogeneous systems.     

对颗粒物质运动的一致性进行控制的随机力场

在颗粒流的研究中引入了一个正态分布的随机力场, 并通过计算机模拟研究了该力场对均匀颗粒流的影响. 结果发现: 随机力场基本上不改变均匀颗粒流的平均密度和速度, 对颗粒流密度的涨落也影响很小. 随机力场对均匀颗粒流的影响主要体现在它可提高速度的涨落, 它与颗粒体系的耗散性质相抗衡, 使颗粒流维持一定的波动能量. 研究结果还显示: 通过随机力场所获得的波动能量并没有均匀分布到各个自由度上, 由于颗粒体系的耗散性质颗粒体系难以达到能均分状态. 关键词： 颗粒物质 随机力 分子动力学模拟